f(n+1)=f(n-1)-f(n)

Solution for f(n+1)=f(n-1)-f(n) equation:

Simplifying
f(n + 1) = f(n + -1) + -1f(n)

Reorder the terms:
f(1 + n) = f(n + -1) + -1f(n)
(1 * f + n * f) = f(n + -1) + -1f(n)
(1f + fn) = f(n + -1) + -1f(n)

Reorder the terms:
1f + fn = f(-1 + n) + -1f(n)
1f + fn = (-1 * f + n * f) + -1f(n)
1f + fn = (-1f + fn) + -1f(n)

Multiply f * n
1f + fn = -1f + fn + -1fn

Combine like terms: fn + -1fn = 0
1f + fn = -1f + 0
1f + fn = -1f

Solving
1f + fn = -1f

Solving for variable 'f'.

Move all terms containing f to the left, all other terms to the right.

Add 'f' to each side of the equation.
1f + f + fn = -1f + f

Combine like terms: 1f + f = 2f
2f + fn = -1f + f

Combine like terms: -1f + f = 0
2f + fn = 0

Factor out the Greatest Common Factor (GCF), 'f'.
f(2 + n) = 0

Subproblem 1
Set the factor 'f' equal to zero and attempt to solve:

Simplifying
f = 0

Solving
f = 0

Move all terms containing f to the left, all other terms to the right.

Simplifying
f = 0
Subproblem 2
Set the factor '(2 + n)' equal to zero and attempt to solve:

Simplifying
2 + n = 0

Solving
2 + n = 0

Move all terms containing f to the left, all other terms to the right.

Add '-2' to each side of the equation.
2 + -2 + n = 0 + -2

Combine like terms: 2 + -2 = 0
0 + n = 0 + -2
n = 0 + -2

Combine like terms: 0 + -2 = -2
n = -2

Add '-1n' to each side of the equation.
n + -1n = -2 + -1n

Combine like terms: n + -1n = 0
0 = -2 + -1n

Simplifying
0 = -2 + -1n

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Solution
f = {0}